Stratification and the smashing spectrum

TL;DR

This paper develops a theory of stratification in tensor-triangulated categories using the smashing spectrum, connecting various prime ideals and exploring conditions for the Telescope Conjecture.

Contribution

It introduces a new framework linking smashing spectra, stratification, and the Telescope Conjecture, with results on descent and spectrum properties.

Findings

Telescope Conjecture holds iff the homological spectrum is T0 and support detects vanishing.

Stratification reduces to smashing localizations.

Smashing spectrum T0 condition characterizes the Telescope Conjecture outside stratified contexts.

Abstract

We develop the theory of stratification for a rigidly-compactly generated tensor-triangulated category using the smashing spectrum and the small smashing support. Within the stratified context, we investigate connections between big prime ideals, objectwise-prime ideals and homological primes, and we show that the Telescope Conjecture holds if and only if the homological spectrum is $T_0$ and the homological support detects vanishing. We also reduce stratification to smashing localizations. Moreover, we study induced maps between smashing spectra and prove a descent theorem for stratification. Outside the stratified context, we prove that the Telescope Conjecture holds if and only if the smashing spectrum is $T_0$ with respect to the small topology.